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### The algorithm for adjacent vertex distinguishing proper edge coloring of graphs

Jingwen Li, Tengyun Hu, Fei Wen
2015 Discrete Mathematics, Algorithms and Applications (DMAA)
An adjacent vertex distinguishing proper edge coloring of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meet the same set of colors.  ...  The minimum number of colors is called adjacent vertex distinguishing proper edge chromatic number of G.  ...  proof on conjecture of adjacent vertex distinguishing proper edge coloring.  ...

### On Twin Edge Colorings of d Infinite Paths

Huan YANG, Shuang-liang TIAN, Su-su JIAO, Xia-hong CAI
2017 DEStech Transactions on Engineering and Technology Research
Let σ be a proper edge coloring of a connected graph G of order at least 3, where the If σ can induce a proper vertex coloring of G , then σ is called a twin edge k-coloring of G .  ...  The minimum number of colors for which G has a twin edge coloring is called the twin chromatic index of G .  ...  Recently, an adjacent vertex distinguishing edge-coloring  of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices has the same set of colors.  ...

### Color-blind index in graphs of very low degree

Jennifer Diemunsch, Nathan Graber, Lucas Kramer, Victor Larsen, Lauren M. Nelsen, Luke L. Nelsen, Devon Sigler, Derrick Stolee, Charlie Suer
2017 Discrete Applied Mathematics
When c^* induces a proper vertex coloring, that is, c^*(u)≠ c^*(v) for every edge uv in G, we say that c is color-blind distinguishing.  ...  Let c:E(G)→ [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̅(v)=(a_1,...,a_k), where a_i is the number of edges incident to v with color i.  ...  The edge-coloring c is neighbor distinguishing ifc is a proper vertex coloring of the vertices of G.  ...

### Concise proofs for adjacent vertex-distinguishing total colorings

Jonathan Hulgan
2009 Discrete Mathematics
We say that f is an adjacent vertex-distinguishing total coloring if for any two adjacent vertices, the set of colors appearing on the vertex and incident edges are different.  ...  Let G = (V , E) be a graph and f : (V ∪E) → [k] be a proper total k-coloring of G.  ...  This is an AVDTC because it is clearly proper and the color sets of adjacent vertices are distinguished by their vertex color.  ...

### A PROPER TOTAL COLORING DISTINGUISHING ADJACENT VERTICES BY SUMS OF SOME PRODUCT GRAPHS

Hana Choi, Dongseok Kim, Sungjin Lee, Yeonhee Lee
2015 Communications of the Korean Mathematical Society
In this article, we consider a proper total coloring distinguishes adjacent vertices by sums, if every two adjacent vertices have different total sum of colors of the edges incident to the vertex and the  ...  color of the vertex.  ...  Woźniak  first introduced that a proper total coloring of Γ is a proper total colorings distinguishing adjacent vertices by sums if for a vertex v ∈ V (Γ), the total sum of colors of the edges incident  ...

### Smarandachely Adjacent Vertex Distinguishing Edge Coloring of Some Graphs

Shunqin Liu
2019 Journal of Physics, Conference Series
A Series of new coloring problems (such as vertex distinguishing edge coloring, vertex distinguishing total coloring, adjacent vertex distinguishing edge coloring, smarandachely adjacent vertex distinguishing  ...  At present, there are relatively few articles on smarandachely adjacent vertex distinguishing edge coloring.  ...  Then a k − proper edge coloring f of a graph G is called a smrandachely adjacent vertex distinguishing edge coloring if for all adjacent vertex u and v , we have ( ) Evidently graph with one-degree -vertex  ...

### Neighbor sum distinguishing total coloring of 2-degenerate graphs

Jingjing Yao, Xiaowei Yu, Guanghui Wang, Changqing Xu
2016 Journal of combinatorial optimization
A proper k-total coloring of G is called neighbor sum distinguishing if f (u) = f (v) for each edge uv ∈ E(G).  ...  Let f (v) denote the sum of the colors on the edges incident with v and the color on vertex v.  ...  A proper k-total coloring φ of G is called a neighbor sum (set) distinguishing k-total coloring if f (u) = f (v) (S(u) = S(v)) for each edge uv ∈ E(G), where f (v) (S(v)) is the sum (set) of colors on  ...

### Optimal Adjacent Vertex-Distinguishing Edge-Colorings of Circulant Graphs [article]

2022 arXiv   pre-print
A k-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two adjacent vertices are distinguished by the set of colors appearing in the edges incident to each vertex.  ...  The smallest value k for which G admits such coloring is denoted by χ'_a(G). We prove that χ'_a(G) = 2R + 1 for most circulant graphs C_n([1, R]).  ...  Some interest has been shown in non-proper adjacent vertex-distinguishing edge-colorings.  ...

### On the Vertex-Distinguishing Edge Chromatic Number of 𝐏𝐦 ∨ 𝐂𝐧

Chuan-cheng ZHAO, Shu-xia YAO, Zhi-guo REN
2017 DEStech Transactions on Engineering and Technology Research
In this paper, we present the edge coloring of join-graphs about path and cycle, and gain the vertex-distinguishing edge chromatic number of ∨ .  ...  A proper edge coloring of graph G is called equitable adjacent strong edge coloring if colored sets from every two adjacent vertices incident edge are different, and the number of edges in any two color  ...  In  introduced the vertex-distinguishing edge coloring of graph, and give the correspondence conjecture.  ...

### On the Total-Neighbor-Distinguishing Index by Sums

Monika Pilśniak, Mariusz Woźniak
2013 Graphs and Combinatorics
We consider a proper coloring c of edges and vertices in a simple graph and the sum f (v) of colors of all the edges incident to v and the color of a vertex v.  ...  We conjecture that + 3 colors suffice to distinguish adjacent vertices in any simple graph.  ...  Recently, some other authors in  considered a proper coloring of edges distinguishing adjacent vertices by sums.  ...

### On the adjacent vertex distinguishing total coloring numbers of graphs with Δ=3

Xiang'en Chen
2008 Discrete Mathematics
An adjacent vertex distinguishing total-coloring of a simple graph G is a proper total-coloring of G such that no pair of adjacent vertices meets the same set of colors.  ...  The minimum number of colors a (G) required to give G an adjacent vertex distinguishing total-coloring is studied. We proved a (G) 6 for graphs with maximum degree (G) = 3 in this paper.  ...  From  we know that the vertex distinguishing proper edge coloring number of G is at most 3s + t + 2.  ...

### L-factors and adjacent vertex-distinguishing edge-weighting [article]

Yinghua Duan, Hongliang Lu, Qinglin yu
2010 arXiv   pre-print
Using these results, we investigate edge-weighting problem. In particular, we prove that every 4-colorable graph admits a vertex-coloring 4-edge-weighting.  ...  An edge weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on edge weighting problem, several types of vertex-coloring problems are put forward.  ...  The coloring is proper if no two adjacent vertices share the same color. A graph G is k-colorable if G has a proper k-vertex coloring.  ...

### Probing Graph Proper Total Colorings With Additional Constrained Conditions [article]

Bing Yao, Ming Yao, Xiang-en Chen
2015 arXiv   pre-print
Graph proper total colorings with additional constrained conditions have been investigated intensively in the last decade year.  ...  In this article some new graph proper total colorings with additional constrained conditions are defined, and approximations to the chromatic numbers of these colorings are researched, as well as some  ...  In the article  , Burris and Schelp introduce that a proper edge-coloring of a simple graph G is called a vertex distinguishing edge-coloring (vdec) if for any two distinct vertices u and v of G, the  ...

### The Vertex-Distinguishing Edge Coloring of 𝑷𝒎⋁𝑲𝒏 and 𝑪𝒎 ∨ 𝑲𝒏

Chuan-cheng ZHAO, Shu-xia YAO, Liu JUN
2017 DEStech Transactions on Engineering and Technology Research
In this paper, we obtain vertex-distinguishing edge coloring of ∨ and ∨ .  ...  A proper edge coloring of graph G is called equitable adjacent strong edge coloring if colored sets from every two adjacent vertices incident edge are different, and the number of edges in any two color  ...  In  introduced the vertex-distinguishing edge coloring of graph, and give the correspondence conjecture.  ...

### A note on the vertex-distinguishing proper coloring of graphs with large minimum degree

Cristina Bazgan, Amel Harkat-Benhamdine, Hao Li, Mariusz Woźniak
2001 Discrete Mathematics
We prove that the number of colors required to properly color the edges of a graph of order n and (G) ¿ n=3 in such a way that any two vertices are incident with di erent sets of colors is at most (G)  ...  The coloring f is proper if no two adjacent edges are assigned the same color and vertex-distinguishing proper (VDP) coloring if it is proper and F(u) = F(v) for any two distinct vertices u; v.  ...  The minimum number of colors required to ÿnd a VDP coloring of a graph G without isolated edges and with at most one isolated vertex is called the vertex-distinguishing proper edge-coloring number and  ...
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